Allison needs 13 inches of cloth for a sewing project, but the fabric store only sells the cloth by the centimeter. Since 1 inch is the same as 2. 54 centimeters, how much cloth will Allison buy in centimeters?

Answer:

y = sine (x + 90 degrees)

Step-by-step explanation:

y = cosine(x) is a curve that crosses the y-axis at y = 1 and completes one cycle at 360 degrees.

sine(x) have the same curve than cosine(x), but translated 90° to the right respect cosine(x)

f(x + c) translates f(x) horizontally c units to the left.

Then, sine(x + 90) is equivalent to cosine(x)

The graph that shows the given transformation is option A.

We can see the graph of f(x) on the top of the given image, and we want to identify the graph of the transformation f(4x), this will be an horizontal compression of the function.

So we should look for a graph that looks like the first one, but its compresed on the horizontal axis. The vertical axis is not changed, remember that.

The only one that shows that is graph A, so that is the correct option.

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Step-by-step explanation:

The volume of each steel rod can be calculated using the formula for the volume of a cylinder:

V = πr^2h

where r is the radius of the cylinder (half the diameter), and h is the height.

In this case, the diameter is 8 cm, so the radius is 4 cm. The height is 20 cm. Therefore, the volume of each steel rod is:

V = π(4 cm)^2(20 cm) = 320π cm^3

To find out how much steel is needed to make 146 rods, we can multiply the volume of one rod by the number of rods:

Total volume = 146 rods x 320π cm^3/rod

Total volume = 46,720π cm^3

We can use a calculator to approximate the value of π to 3.14 and then calculate the total volume:

Total volume ≈ 46,720 x 3.14 cm^3

Total volume ≈ 146,748.8 cm^3

Rounding to the nearest hundredth, the company will need approximately 146,748.8 cubic centimeters of steel to make 146 rods.

In order to test for any positive autocorrelation in the data, you would use the Pearson correlation coefficient. This measures the strength and direction of a linear relationship between two variables. The level of significance that you set depends on the size of your dataset, but generally should be 0.05 or less.

To test for positive autocorrelation, you need to compare the Pearson correlation coefficient with the critical value of the correlation coefficient. The critical value is determined by the level of significance you have set. If the Pearson correlation coefficient is greater than the critical value, then you can conclude that there is a significant positive autocorrelation.

It is important to keep in mind that a positive correlation does not necessarily mean that there is a cause and effect relationship between the two variables. It only means that the two variables move in the same direction.

In conclusion, if the Pearson correlation coefficient is greater than the critical value, then you can conclude that there is a significant positive autocorrelation.

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The mean score is 96.

The variance is 160.

The standard deviation is approximately 12.65.

To find the mean (average) of the scores, we add them up and divide by the total number of scores:

Mean = (101 + 115 + 78 + 87 + 99) / 5 = 96

So the mean score is 96.

We deduct the lowest score from the highest score to determine the range:

Range = highest score - lowest score = 115 - 78 = 37

So the range is 37.

To find the variance, we need to find the squared difference between each score and the mean, then add up all those squared differences and divide by the number of scores.

The formula for variance is:

Variance =   [tex]\frac{\sum(x_i - \bar x)^2}{n}[/tex]

where

Σ represents the sum of,

[tex]x_i[/tex] is the ith score,

n is the number of scores,

[tex]\bar x[/tex] is the mean.

Using the formula, we get:

Variance = [(101 - 96)² + (115 - 96)² + (78 - 96)² + (87 - 96)² + (99 - 96)²] / 5

= (25 + 361 + 324 + 81 + 9) / 5

= 800 / 5

= 160

So the variance is 160.

We take the square root of the variance to calculate the standard deviation:

Standard deviation = √160 ≈ 12.65

So the standard deviation is approximately 12.65.

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The density of the lead cuboid is approximately 11.3 g/cm³.

What is density?

Density is a physical property of matter that describes how much mass is contained within a certain volume. It is defined as the amount of mass per unit of volume, and is typically measured in grams per cubic centimeter (g/cm³) or kilograms per cubic meter (kg/m³).

The formula for density is:

density = mass / volume

where mass is the amount of matter in an object, and volume is the amount of space that the object occupies.

The volume of the cuboid can be calculated using the formula V = lwh, where l, w, and h are the length, width, and height of the cuboid, respectively.

Given that the length (l) of the cuboid is 5cm, the width (w) is 3cm, and the height (h) is 2cm, we can calculate the volume as:

V = lwh = 5cm x 3cm x 2cm = 30 cm³

The density (ρ) of an object is defined as its mass (m) per unit volume (V), or ρ = m/V.

Given that the mass of the cuboid is 339 g and its volume is 30 cm³, we can calculate the density as:

ρ = m/V = 339 g / 30 cm³ ≈ 11.3 g/cm³ (to 1 decimal place)

Therefore, the density of the lead cuboid is approximately 11.3 g/cm³.

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Answer:

x = -12 and x = 12.

Step-by-step explanation:

To solve for x, we need to take the square root of both sides of the equation:

x² = 144

√(x²) = √144

since sqaure root gives both positive and neagtive value, so

x = ±12

So the solutions for x are x = -12 and x = 12.

If The ratio of the mass of the fruit to the mass of the yoghurt is 2:3 the mass of the yogurt is 99 g,

Let's assume the mass of the fruit is 2x and the mass of the yogurt is 3x, where x is a common factor.

According to the problem, the total mess of the desert is 165 g, so we can write an equation:

2x + 3x = 165

Simplifying this equation, we get:

5x = 165

x = 33

Therefore, the mass of the fruit is:

2x = 2(33) = 66 g

And the mass of the yogurt is:

3x = 3(33) = 99 g

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Answer:

x ≥ 4

Step-by-step explanation:

The mean of the sampling distribution of p is 0.40 and the standard deviation of the sampling distribution of p is 0.0409.

Standard deviation is a measure of how spread out the values in a data set are from the mean. It is calculated by taking the square root of the variance of the data set. It is a commonly used measure to assess the variability in a data set.

The mean of the sampling distribution of p is 0.40, which is the same as the proportion of supporters in the population of Okeechobee County. This is due to the fact that the sample proportion estimates the population proportion. The standard deviation of the sampling distribution of p is equal to the square root of the product of p (0.40) and q (1-p, or 0.60) divided by the sample size (135). In this case, the standard deviation of the sampling distribution of p is 0.0409.

The mean and standard deviation of the sampling distribution of p can be used to understand what values of p are likely when samples of size n = 135 are drawn from the population of Okeechobee County. If a sample is drawn from the population, the sample proportion of supporters is likely to be close to 0.40, which is the mean of the sampling distribution of p. Furthermore, most of the sample proportions of supporters are likely to be within two standard deviations (or 0.0818) of the mean. This means that the sample proportion of supporters is likely to be between 0.3182 and 0.4818 when samples of size n = 135 are drawn from the population.

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The acceleration of a bus his speech changes from 126M/S to 3097 m/s over a period of three seconds is 990. 3 m/s^2

Acceleration is can be defined as the rate of change of the velocity of an object with respect to time.

It is also defined as the rate at which velocity changes with time, that is, in both speed and direction.

Accelerations are vector quantities because they are known to have both magnitude and direction.

The orientation of an object's acceleration is determined by the configuration of the net force acting on that object.

The formula for acceleration is given as;

Acceleration = change in velocity/time

Substitute the values

Acceleration = 3097 - 126 (m/s)/3 s

Divide the values

Acceleration = 990. 3 m/s^2

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Answer:

3

Step-by-step explanation:

in a three - dimensional scenario a plane consists of three points and also they are not on the same line.

Option (a) [tex]y=\frac{3x}{2} -5[/tex] can be used as a the model line on the graph for the Linear function.

A linear function in mathematics is one that has either one or two variables but no exponents. It is a function with a straight line as its graph.

A straight line on the coordinate plane is described by a linear function.

Option (a) [tex]y=\frac{3x}{2} -5[/tex] satisfy with all point as below,

If we take, x = 0, we get y = -5

If we take, x = 2, we get y = -2

If we take, x = 4, we get y = 1

If we take, x = 6, we get y = 4

If we take, x = 8, we get y = 7

So, here we see all the value are match according to the graph.

Therefore, [tex]y=\frac{3x}{2} -5[/tex] can be used as a the model line on the graph for the Linear function.

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Cοnclusiοn that can be drawn frοm the circle graph is 'Pοp is the preferred music fοr 140 students.'

Cοrrect οptiοn is A.

In math, a percentage is a number οr a ratiο that can be expressed as a fractiοn οf 100. If we need tο calculate the percentage οf a number, divide that number by whοle and multiply by 100. Therefοre, percentage means οne part per hundred.

Given,

Tοtal number οf students = 400

By circle graph five sectiοns labeled rοck 13 percent, hip hοp 25 percent, pοp 35 percent, classical 13 percent, and jazz 14 percent

Number οf students preferring pοp music

= 35%

= (35/100) × 400

= 35 × 4

= 140

Number οf students preferring classical music

= 13%

= (13/100) × 400

= 13 × 4

= 52

Number οf students preferring jazz music

= 14%

= (14/100) × 400

= 14 × 4

= 56

Number οf students οf pοp and classical tοgether = 140 + 52 = 192

Number οf students οf classical and jazz tοgether = 52 + 56 = 108

Hence, the οnly cοnclusiοn that is true is, Pοp is the preferred music fοr 140 students.

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Answer:R=3\\

(a,b)=(1,2)

Step-by-step explanation:

[tex](x-a)^{2}+(y-b)^{2}=R^{2}\\(a,b)-center\\R-radius\\x^{2} + 2x - 1 + y^{2}+ 4y = 3\\(x+1)^2=x^{2}+2x+1\\(y+2)^{2}=y^{2}+ 4y+4\\x^{2} + 2x - 1 + y^{2}+ 4y = (x+1)^2-1+(y+2)^2-4-1=(x+1)^2+(y+2)^2-6=3\\(x+1)^2+(y+2)^2=9\\R=3\\(a,b)=(1,2)[/tex]

The temperature in degrees Fahrenheit is approximately 77.4°F.

There are two widely used temperature scales in the world: Celsius and Fahrenheit. The freezing and boiling temperatures of water serve as the foundation for the Celsius scale, commonly referred to as the centigrade scale. According to this scale, the freezing point of water is 0°C, while the boiling point is 100°C.

On the other hand, a German physicist by the name of Daniel Gabriel Fahrenheit created the Fahrenheit scale. While it makes use of different reference points, this scale is similarly based on the freezing and boiling points of water.

The formula of the conversion is given by equation:

25 = 559/9 * (F - 32).

We can solve for F as follows:

25 = 559/9 * (F - 32)

25 * 9/559 = F - 32

F = 25 * 9/559 + 32

F ≈ 77.4

Therefore, the temperature in degrees Fahrenheit is approximately 77.4°F.

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The algebraic expression is simplified to 13. Option A

These are expressions that are made up of terms, variables, constants and coefficients.

They are also made up of arithmetic operations like subtraction, division, multiplication, bracket, and parentheses.

The square root of a number is defined as the factor that when multiplied by itself gives the original number.

The symbol for square root is '√'. The square root of a number is the opposite of squaring a number.

From the information given, we have that;

√169

Now, let's determine the square root

13

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The equation that represents the maximum area enclosed by the fence is 600x - 2x².

What is the maximum area?

Any rectangle's maximal surface area requires a minimum length-to-breadth ratio. Hence, in this scenario, the length must equal the ceiling (perimeter / 4) and the width, of the floor (perimeter /4). Therefore a rectangle with a given perimeter has a maximum area equal to the ceiling(perimeter/4) and the floor(perimeter/4)

Here, we have

Given: A rectangular field bordering a river is to be enclosed with 600 m of fencing. No fence is needed along the riverbank.

We have to find the equation that represents the maximum area enclosed by the fence.

The sum of the sides NOT along the river x + x + y = 600,  and the area equals xy.

This makes the two equations:    2x + y = 600,   and A = xy.

To find the largest area, we need to find A as a function of x or y.  I suggest solving the first equation for y and replacing that in the second equation.

y = 600 - 2x.    and A(x) = x(600-2x)

We now need to maximize A(x) = 600x - 2x²

Hence, the equation that represents the maximum area enclosed by the fence is 600x - 2x².

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in essence, which of the two fractions is the larger one, well, let's put both with the same denominator by multiplying each other by the other's denominator.

[tex]12:15\qquad 9:10\hspace{5em}\cfrac{12}{15}\qquad \cfrac{9}{10} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{12}{15}\cdot \cfrac{10}{10}\implies \cfrac{120}{150} ~~ \bigotimes\hspace{9em}\cfrac{9}{10}\cdot \cfrac{15}{15}\implies \cfrac{135}{150} ~~ \textit{\LARGE \checkmark}[/tex]

Answer:9/10

Step-by-step explanation:

90% average vs 80% average

The square based pyramid has a surface area of is 224 mm²

Given that

A square based pyramid

The shapes in the net of a square based pyramid are

The shape's net is attached

To determine the surface area, we use

Area = l² + 2hl

Where

l = length of the square base = 8 mm

h = slant height = 10 mm

By substitution, we have

Area = 8² + 2 * 10 * 8

Evaluate

Area = 224

Hence, the surface area is 224 mm²

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The side of the right triangle can be found as follows:

b = 4√2 units

d = 7√3 / 2 units

A right triangle is a triangle that has one of its angles as 90 degrees. The sum of angles in a triangle is 180 degrees.

Therefore, let's use trigonometric ratios to find the side of the right triangle as follows;

Let's find b as follows:

sin 45 = opposite / hypotenuse

sin 45° = b / 8

cross multiply

b = 8 sin 45

b = 8 × √2 / 2

b = 8√2 / 2

b = 4√2 units

Let's find d as follows:

sin 60° = d / 7

cross multiply

d = 7 sin 60

d = 7 × √3 / 2

d = 7√3 / 2 units

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Sensitivity 67% specificity 86%

Positive predictive values 67% Negative predictive values 86%

Probability of the correct diagnosis 80%

The following can be calculated from the data given:1. Sensitivity and Specificity2. Positive and Negative Predictive Values.3. The probability of the correct diagnosis1. Sensitivity and Specificity:Sensitivity is the proportion of positive cases correctly identified by a screening test or diagnosis test. Sensitivity = Number of true positives / (Number of true positives + Number of false negatives) = 20 / (20 + 10) = 20 / 30 = 67%Specificity is the proportion of negative cases correctly identified by a screening test or diagnosis test. Specificity = Number of true negatives / (Number of true negatives + Number of false positives) = 60 / (60 + 10) = 60 / 70 = 86%2. Positive and Negative Predictive Values.Positive Predictive Value (PPV) is the proportion of positive test results that are true positives.Positive predictive value (PPV) = Number of true positives / (Number of true positives + Number of false positives) = 20 / (20 + 10) = 20 / 30 = 67%.Negative Predictive Value (NPV) is the proportion of negative test results that are true negatives.Negative predictive value (NPV) = Number of true negatives / (Number of true negatives + Number of false negatives) = 60 / (60 + 10) = 60 / 70 = 86%.3. Probability of the correct diagnosisThe probability of correctly diagnosing a diseased individual would be the proportion of true positives and true negatives among the total tested.Probability of correctly diagnosing a diseased individual = (Number of true positives + Number of true negatives) / (Number of true positives + Number of false positives + Number of true negatives + Number of false negatives) = (20 + 60) / (20 + 10 + 60 + 10) = 80 / 100 = 80%.

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The 5 number summary for the given dataset (40, 42, 46, 48, 51, 55, 58, 66, 67, 68, 69) is: Minimum: 40, Q1: 47, Median (Q2): 55, Q3: 66.5, Maximum: 69.

What is statistics ?

Statistics is a branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of data. It involves the use of quantitative methods to gather, describe, and draw inferences from data, which can be used to make decisions, predictions, or conclusions about a population or a phenomenon.

To find the 5 number summary, we need to find the minimum, maximum, and the three quartiles of the given dataset:

The minimum value is 40.

To find the quartiles, we first need to sort the data in ascending order:

40, 42, 46, 48, 51, 55, 58, 66, 67, 68, 69

The median (quartile 2) is the middle value of the dataset, which is 55.

Quartile 1 (Q1) is the median of the lower half of the dataset, which includes the values 40, 42, 46, 48, and 51. The median of this lower half is (46+48)/2 = 47.

Quartile 3 (Q3) is the median of the upper half of the dataset, which includes the values 58, 66, 67, 68, and 69. The median of this upper half is (66+67)/2 = 66.5.

The maximum value is 69.

Therefore, the 5 number summary for the given dataset is:

Minimum: 40

Q1: 47

Median (Q2): 55

Q3: 66.5

Maximum: 69

Therefore, the 5 number summary for the given dataset (40, 42, 46, 48, 51, 55, 58, 66, 67, 68, 69) is: Minimum: 40, Q1: 47, Median (Q2): 55, Q3: 66.5, Maximum: 69.

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The tax rate used to calculate a person's payable tax in South Africa is not a fixed rate, but rather a combination of different rates that apply to different income brackets

As of the 2021/2022 tax year, the tax rates for individuals in South Africa are as follows:

For taxable income up to R216,200: 18% of taxable income

For taxable income between R216,201 and R337,800: R38,916 plus 26% of taxable income above R216,200

For taxable income between R337,801 and R467,500: R70,532 plus 31% of taxable income above R337,800

For taxable income between R467,501 and R613,600: R110,739 plus 36% of taxable income above R467,500

For taxable income between R613,601 and R782,200: R163,335 plus 39% of taxable income above R613,600

For taxable income above R782,200: R229,089 plus 41% of taxable income above R782,200

Based on the above tax rates, if a person's annual taxable income is R450,090, their payable tax would be calculated as follows:

18% of the first R216,200 = R38,916

26% of the amount above R216,200 up to R337,800 = R26,890

31% of the amount above R337,800 up to R450,090 = R34,754

Total payable tax = R38,916 + R26,890 + R34,754 = R100,560

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The value of the n for the given algebraic equation is=12.

The concept of algebraic expressions is the use of letters or alphabets to represent numbers without providing their precise values. We learned how to express an unknown value using letters like x, y, and z in the fundamentals of algebra. Here, we refer to these letters as variables. Variables and constants can both be used in an algebraic expression. A coefficient is any value that is added to a variable before being multiplied by it.

The given algebraic expression is

-4.3+1.2n=10.1

Add 4.3 on both sides

1.2n=10.1+4.3

1.2n=14.4

divide 1.2 into both sides

n = 14.4÷1.2

n = 12

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Answer: a) To find the scale factor, we need to find the ratio of the corresponding lengths of the two prisms, since the volumes of two similar prisms are proportional to the cube of the scale factor. Let x be the scale factor.

(volume of larger prism) / (volume of smaller prism) = (x³) = 27/1

x³ = 27

x = 3

Therefore, the scale factor is 3.

b) The ratio of the areas is equal to the square of the scale factor. Let A1 and A2 be the areas of the larger and smaller prisms, respectively.

(A1) / (A2) = (scale factor)²

A2 = A1 / (scale factor)²

A2 = 153 / 9

A2 = 17 cm²

Therefore, the ratio of their areas is 9:1.

c) The ratio of the volumes is equal to the cube of the scale factor.

(volume of larger prism) / (volume of smaller prism) = (scale factor)³

(volume of smaller prism) = (1 cm³) * (scale factor)³

(volume of smaller prism) = 27 cm³

Therefore, the ratio of their volumes is 27:1.

d) To find the surface area of the smaller prism, we can use the same ratio as in part (b) to find that the surface area of the smaller prism is equal to the surface area of the larger prism divided by the square of the scale factor.

(surface area of smaller prism) = (surface area of larger prism) / (scale factor)²

(surface area of smaller prism) = 153 / 9

(surface area of smaller prism) = 17 cm²

Therefore, the surface area of the smaller prism is 17 cm².

Step-by-step explanation:

The equal monthly payments for a car costs $22,000 with a down payment of $4,000 based on each scenario are:

Let's discuss each scenario we have.

5. For calculating the amount of each payment, the first thing to do is to find the balance after the down payment is made.

The amount of the down payment is $4,000.

The total cost of the car is $22,000.

Hence the balance is:

Balance = Total cost of the car − Down payment

Balance = $22,000 − $4,000 = $18,000

Now we can calculate the amount of each payment.

The payment is to be made in 48 equal monthly installments.

The interest rate is 12% per year.

We need to convert this into the monthly interest rate by dividing by 12.

Hence:

Monthly interest rate = 12% / 12 = 1% = 0.01

Let x be the amount of each monthly payment. Then the equation that represents the above conditions is given by:

18000=x(1−(1+0.01)−48)/0.01

Using the formula (1−(1+ r/n)−nt) = (1−(1+r)^−n), we can simplify the above equation to:

18000=x(1−(1+0.01)−48)/0.01x

=18000/28.95x

≈621.92

Hence the amount of each equal monthly payment is approximately $621.92.

6. For calculating the amount of each payment, the first thing to do is to find the balance after the down payment is made.

The amount of the down payment is $4,000.

The total cost of the car is $22,000.

Hence the balance is:

Balance = Total cost of the car − Down payment

Balance = $22,000 − $4,000 = $18,000

Now we can calculate the amount of each payment.

The payment is to be made in 60 equal monthly installments.

The interest rate is 12% per year.

We need to convert this into the monthly interest rate by dividing by 12.

Hence:

Monthly interest rate = 12% / 12 = 1% = 0.01

Let x be the amount of each monthly payment.

Then the equation that represents the above conditions is given by: 18000=x(1−(1+0.01)−60)/0.01

Using the formula (1−(1+ r/n)−nt) = (1−(1+r)^−n), we can simplify the above equation to:

18000=x(1−(1+0.01)−60)/0.01x

=18000/35.643x

≈505.01

Hence the amount of each payment is approximately $505.01.

7. For calculating the amount of each payment, the first thing to do is to find the balance after the down payment is made.

The amount of the down payment is $4,000.

The total cost of the car is $22,000.

Hence the balance is:

Balance = Total cost of the car − Down payment

Balance = $22,000 − $4,000 = $18,000

Now we can calculate the amount of each payment.

The payment is to be made in 72 equal monthly installments.

The interest rate is 12% per year.

We need to convert this into the monthly interest rate by dividing by 12.

Hence:

Monthly interest rate = 12% / 12 = 1% = 0.01

Let x be the amount of each monthly payment.

Then the equation that represents the above conditions is given by:

18000=x(1−(1+0.01)−72)/0.01

Using the formula (1−(1+ r/n)−nt) = (1−(1+r)^−n), we can simplify the above equation to:

18000=x(1−(1+0.01)−72)/0.01x

=18000/46.869x

≈383.63

Hence the amount of each payment is approximately $383.63.

8. For calculating the amount of each payment, the first thing to do is to find the balance after the down payment is made.

The amount of the down payment is $4,000.

The total cost of the car is $22,000.

Hence the balance is:

Balance = Total cost of the car − Down payment

Balance = $22,000 − $4,000 = $18,000

Now we can calculate the amount of each payment.

The payment is to be made in 48 equal monthly installments.

The interest rate is 18% per year.

We need to convert this into the monthly interest rate by dividing by 12.

Hence:

Monthly interest rate = 18% / 12 = 1.5% = 0.015

Let x be the amount of each monthly payment.

Then the equation that represents the above conditions is given by:

18000=x(1−(1+0.015)−48)/0.015

Using the formula (1−(1+ r/n)−nt) = (1−(1+r)^−n), we can simplify the above equation to:

18000=x(1−(1+0.015)−48)/0.015x

=18000/25.525x

≈704.26

Hence the amount of each payment is approximately $704.26.

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Answer:

  2520 m²

Step-by-step explanation:

You want the surface area of the irregular prism shown.

The perimeter of the base (shaded front face) is the sum of the lengths of the horizontal edges and the lengths of the vertical edges. That sum is ...

  2(20 m) +2(20 m) = 80 m

The lateral area of the prism is the sum of the unshaded rectangle areas. That total area is the product of the perimeter of the base and the distance between bases:

  LA = Ph = (80 m)(25 m) = 2000 m²

The area of each of the two bases can be found several ways. We find the height of the middle step to be (20 m -7 m -6 m) = 7 m, so the bottom step fits nicely into the space not used by the middle step. (This is shown in the attachment.)

That is, the base area can be computed as the area of a rectangle 13 m high and 20 m wide:

  Base area = WH = (20 m)(13 m) = 260 m²

The total surface area of the prism is the sum of the lateral area and the area of the two bases:

  SA = LA +2·B = (2000 m²) +2(260 m²) = 2520 m²

The surface area of the prism is 2520 square meters.

The correct option is ([tex]H) i.e. a² > b².[/tex]

What is an inequality?

In mathematics, an inequality is a statement that one quantity is less than or greater than another quantity. Inequalities are used to compare values and express relationships between them.

For all integers a < 0 and b < 0, the following inequality must be true:

[tex]H. a² > b²[/tex]

This can be proved as follows:

If a and b are both negative, then a² and b² are both positive. Since a < 0 and b < 0, we have a < b or b < a, which implies that[tex]a² > b² or b² > a².[/tex]

If a and b are both negative and equal (i.e., a = b < 0), then a² = b², which implies that a² > b² and b² > a² are both false.

None of the other inequalities (F, G, J, K) must be true for all such a and b. For example:

F. a/b > 1 is not necessarily true, since a and b could both be negative and have the same absolute value (e.g., a = b = -1), in which case a/b = 1.

G. ab > 2 is not necessarily true, since a and b could both be negative and have absolute values less than 2 (e.g., a = b = -1), in which case ab = 1 < 2.

J. [tex]a²b³ > 0 is[/tex] not necessarily true, since a and b could both be negative and have different absolute values (e.g., a = -2 and b = -1), in which case[tex]a²b³ = -8 < 0.[/tex]

K. [tex]a^{4} b² > 0[/tex] is not necessarily true, since a and b could both be negative and have the same absolute value (e.g., a = b = -1), in which case [tex]a^{4} b² = 1 > 0[/tex]. However, if we add the condition that a and b have different absolute values, then K would be true, since [tex]a^{4} b²= (|a|)² |b|² > 0.[/tex]

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The equivalent ratio to tan F  in  right angles triangle, △EFD is tan F = DE / DF.

For the stated question:

△EFD∼△GHD

So, in right angles triangle, △EFD

Sin Ф = perpendicular / hypotenuse.

Sin E = DF / EF

And,

The ratio is equivalent to tan F = perpendicular / base

tan F = DE / DF

Thus, the equivalent ratio to tan F  in  right angles triangle, △EFD is tan F = DE / DF.

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